The fixed time-step variational integrators derived from the discrete variational principle cannot preserve the energy of the system exactly. To conserve the energy in addition to preserving the symplectic structure and conserving the momentum, time adaptation needs to be used. These energy-preserving, adaptive time-step variational integrators were first developed for conservative systems in [1]} by imposing an additional energy preservation equation to compute the time-step. The same integrators were derived through a variational approach for a more general case of time-dependent Lagrangian systems in [2]} by preserving the discrete energy obtained from the discrete variational principle in the extended phase space. Recently, Sharma et al. extended these methods to systems with external nonconservative forcing [3]} and rigid body motion [4]}, [5]}.
